Load cell

ABSTRACT

A load cell combines the outputs of a plurality of strain gauges to measure components of an applied load. Combination of strain gauge outputs allows measurement of any of six load components without requiring complex machining or mechanical linkages to isolate load components. An example six axis load cell produces six independent analog outputs which can be combined to determine any one of the six general load components.

This invention was made with Government support under ContractDE-AC04-94AL85000 awarded by the U.S. Department of Energy. TheGovernment has certain rights in the invention.

BACKGROUND OF THE INVENTION

This invention relates to the field of load cells, specifically loadcells that measure torsion or shear.

Load cells are used for measuring forces and moments along certaindirections. Measurement of loads and moments about multiple axes can bebeneficial in various research and manufacturing applications. Currentmulti-axis load cells, however, require complex machining and mechanicallinkages to isolate loads along multiple axes. Loads that manifest onlythrough shear stress at the load cell typically require especiallycomplex machining or linkages for isolation. See, e.g., Meyer et al.,U.S. Pat. No. 4,640,138; Meyer et al., U.S. Pat. No. 5,315,882; Mullin,U.S. Pat. No. 5,339,697; Rieck et al., U.S. Pat. No. 4,259,863; Ruoff,Jr. et al., U.S. Pat. No. 4,138,884. Current multi-axis load cells areconsequently expensive to manufacture, and can be readily damaged byoverloading. The frequency response of current load cells is alsolimited by the characteristics of the machining and mechanical linkages,precluding their use in applications with rapidly varying loads.

Many applications that might benefit from multi-axis load measurementsare precluded by the high cost of current multi-axis load cells. Also,applications such as many robotics applications encounter widely varyingloads. Some encounter unknown loading, making load cell damage due tooverloading likely. Also, existing load cells can not be integrated intoa robot link, complicating the robotic system.

There is a need for a multi-axis load cell that is simple andinexpensive to manufacture, that is unlikely to be damaged by widelyvarying loads, and that can be readily incorporated into a roboticsystem.

SUMMARY OF THE INVENTION

The present invention provides an improved load cell that is simple andinexpensive to manufacture, and that is capable of measuring a widerange of loads. The present invention uses strain gauges, mounted innovel arrangements on free surfaces of a load cell body, to measureloads.

Generally, six load components are of interest: axial load, axial moment(torsion), two loads at angles to the axis (typically along two axesmutually orthogonal and orthogonal to the cell axis), and moments abouttwo axes at angles to the cell's axis (typically about two axes mutuallyorthogonal and orthogonal to the cell axis). Strain gauges measure axialsurface strain along a given axis, and accordingly do not directlymeasure any of the six loads of interest. Placing sets of strain gaugesat certain locations on free surfaces of the cell, however, can allowthe gauge outputs to be combined to measure any of the six loadcomponents of interest individually. Specific placements of the straingauges, described herein, can provide coupled, linearly-independentrelationships among strain gauge outputs and load components. Individualload components can be determined from the strain gauge outputs and thecoupled linearly-independent relationships.

Advantages and novel features will become apparent to those skilled inthe art upon examination of the following description or may be learnedby practice of the invention. The objects and advantages of theinvention may be realized and attained by means of the instrumentalitiesand combinations particularly pointed out in the appended claims.

DESCRIPTION OF THE FIGURES

The accompanying drawings, which are incorporated into and form part ofthe specification, illustrate embodiments of the invention and, togetherwith the description, serve to explain the principles of the invention.

FIG. 1 is an illustration of an application of a multi-axis load cell.

FIG. 2 is an illustration of general loading of a multi-axis load cell.

DETAILED DESCRIPTION OF THE INVENTION

The present invention provides an improved load cell that is simple andinexpensive to manufacture, and that is capable of measuring a widerange of loads. The present invention uses strain gauges, mounted innovel arrangements on free surfaces of a load cell body, to measureloads.

FIG. 1 illustrates an example application of a load cell according tothe present invention. A robot 11 operates on a workpiece 10. Controller12 monitors and controls the operation of robot 11. Load cell 14provides information via connection 15 to controller 12 concerningforces and moments exerted by or on robot 1 1. For example, load cell 14can provide measurement of axial moment (or torsion) to allow controller12 to direct robot 11 to apply a certain torque to a fastener. Asanother example, load cell 14 can provide measurement of general loadingto allow controller to direct robot 11 to apply selected force toworkpiece 10 during a finishing operation. As another example, load cell14 can provide measurement of general loading to allow controller 12 tosafely control robot 11 when unexpected high loads are encountered (aswould be caused by collisions between robot 11 and objects unexpectedlywithin the workspace of robot 11).

FIG. 2 illustrates general loading of a load cell. Load cell 21 isloaded at ends 22, 23. A coordinate system comprising three axes x, y, zis conventionally used to describe loading of the cell. A first axis zlies substantially parallel to the longitudinal axis of the cell 21. Theother two axes x, y are mutually orthogonal, and are both orthogonal tothe first axis z. General loading of cell 21 comprises force loadcomponents Px, Py, Pz along each of the three axes x, y, z and momentload components Mx, My, Mz about each of the three axes x, y, z. Momentload component Mz about the first axis z is often referred to astorsion. Force load component Pz along the first axis z is oftenreferred to as axial load.

The present invention comprises strain gauges mounted at specifiedlocations and specified orientations on surfaces of a load cell body,where the load cell body material is characterized by its modulus ofelasticity (E) and Poisson's ratio (v). Strain gauges mounted atlocations and orientations according to the present invention allowdetermination of a selected load component without requiring specialmechanical structures or a multiplicity of strain gauges.

Relationships among gauge locations and orientations and sensitivity toparticular load components useful in understanding the present inventionare discussed in Spletzer, U.S. Pat. No. 5,850,044 (Dec. 1998),incorporated herein by reference. Spletzer presents example designs thatprovide signals indicative of individual load components. The presentinvention comprises designs that provide coupled, linearly-independentoutputs, allowing determination of load components from decoupling ofoutput from a minimal number of gauges.

The output of a strain gauge mounted with a surface of a body under loadcan be affected by numerous components of the load. Each gauge outputcan be expressed as a combination of the load components (a load-strainequation). Mounting several gauges with the body yields several gaugeoutputs, each indicative of a combination of load components. Asdescribed below, individual load components can be determined from theseveral gauge outputs as long as the gauge placement and orientationproduces sufficient linearly-independent load-strain equations todetermine the load component of interest.

Certain gauge orientations can result in gauge outputs that areinsensitive to certain load components: gauges at 0 and 90 degrees aresensitive only to bending and axial load; gauges at ±β_(τ) are sensitiveto shear and torsion only. Accordingly, at least three gauges must be atangles other than 0 or 90 degrees if shear or torsion are to bedetermined. Similarly, at least three gauges must be at angles otherthan ±β_(τ) if bending or axial load are to be determined. Further,there must be at least as many linearly-independent load-strainequations as there are load components that influence the gauges'outputs (e.g., gauges with the same load-strain equation as othergauges, or as linear combinations of other equations or gauges, do notprovide new information to the determination of load components).

SIX AXIS, CALIBRATED LOAD CELL

Six strain gauges can be mounted with a body at various orientations andazimuths, subject to the previous constraints. The strain gauge outputscan be calibrated, either by computational simulation or by experiment,to produce a sensitivity matrix relating the strain gauge outputs to theapplied load components. The inverse of the sensitivity matrix decouplesthe load-strain equations, allowing unknown components of applied loadsto be determined from the strain gauge outputs and the inverse of thesensitivity matrix.

SIX AXIS, USING SPECIFIC GAUGE ORIENTATIONS

Mounting gauges at 0 or 90 degrees and at ±β_(τ) can simplify thedetermination of load components since gauges at those angles aresensitive to only certain load components. The general relationship fora gauge at 0 degrees and at azimuth θ is given by equation 1.$\begin{matrix}{ɛ_{0} = \left( {\frac{{M_{x}r\quad \sin \quad \theta} - {M_{y}r\quad \cos \quad \theta}}{EI} + \frac{P_{z}}{EA}} \right)} & (1)\end{matrix}$

The general relationship for a gauge at β_(τ) and at azimuth θ is givenby equation 2. $\begin{matrix}{ɛ_{\beta_{\tau}} = {2\sqrt{v}\left( {\frac{\left( {{{- P_{x}}\sin \quad \theta} + {P_{y}\cos \quad \theta}} \right)Q_{\max}}{{EI}\left( {D - d} \right)} + \frac{M_{z}r}{EJ}} \right)}} & (2)\end{matrix}$

For three gauges placed at arbitrary azimuths θ₁, θ₂, and θ₃, two setsof three coupled equations results between three of the applied loadsand the three strains at either gauge angle. These relations are shownin equations 3 and 4, with the arguments of the strain valuerepresenting the gauge angle and the azimuth. $\begin{matrix}{\begin{bmatrix}\begin{matrix}{ɛ\left( {0,\theta_{1}} \right)} \\{ɛ\left( {0,\theta_{2}} \right)}\end{matrix} \\{ɛ\left( {0,\theta_{3}} \right)}\end{bmatrix} = {\begin{bmatrix}{\sin \quad \theta_{1}} & {\cos \quad \theta_{1}} & 1 \\{\sin \quad \theta_{2}} & {\cos \quad \theta_{2}} & 1 \\{\sin \quad \theta_{3}} & {\cos \quad \theta_{3}} & 1\end{bmatrix}\begin{bmatrix}\begin{matrix}\frac{M_{x}r}{EI} \\{- \frac{M_{y}r}{EI}}\end{matrix} \\\frac{P_{z}}{EA}\end{bmatrix}}} & (3) \\{\begin{bmatrix}\begin{matrix}{ɛ\left( {\beta_{\tau},\theta_{1}} \right)} \\{ɛ\left( {\beta_{\tau},\theta_{2}} \right)}\end{matrix} \\{ɛ\left( {\beta_{\tau},\theta_{3}} \right)}\end{bmatrix} = {\begin{bmatrix}{\sin \quad \theta_{1}} & {\cos \quad \theta_{1}} & 1 \\{\sin \quad \theta_{2}} & {\cos \quad \theta_{2}} & 1 \\{\sin \quad \theta_{3}} & {\cos \quad \theta_{3}} & 1\end{bmatrix}\begin{bmatrix}\begin{matrix}{{- 2}\sqrt{v}\frac{P_{x}Q_{\max}}{E\left( {D - d} \right)}} \\{2\sqrt{v}\frac{P_{y}Q_{\max}}{E\left( {D - d} \right)}}\end{matrix} \\{2\sqrt{v}\frac{M_{z}r}{EJ}}\end{bmatrix}}} & (4)\end{matrix}$

The closed form expressions for each of the six loads i n terms of themeasured strained can be determined by the inverted coefficient matrixfrom equations 3 and 4, as in equation 5. $\begin{matrix}{\begin{bmatrix}{\sin \quad \theta_{1}} & {\cos \quad \theta_{1}} & 1 \\{\sin \quad \theta_{2}} & {\cos \quad \theta_{2}} & 1 \\{\sin \quad \theta_{3}} & {\cos \quad \theta_{3}} & 1\end{bmatrix}^{- 1} = {\frac{\begin{bmatrix}{{\cos \quad \theta_{2}} - {\cos \quad \theta_{3}}} & {{\cos \quad \theta_{3}} - {\cos \quad \theta_{1}}} & {{\cos \quad \theta_{1}} - {\cos \quad \theta_{2}}} \\{{\sin \quad \theta_{3}} - {\sin \quad \theta_{2}}} & {{\sin \quad \theta_{1}} - {\sin \quad \theta_{3}}} & {{\sin \quad \theta_{2}} - {\sin \quad \theta_{1}}} \\{\sin \left( {\theta_{2} - \theta_{3}} \right)} & {\sin \left( {\theta_{3} - \theta_{1}} \right)} & {\sin \left( {\theta_{1} - \theta_{2}} \right)}\end{bmatrix}}{{\sin \left( {\theta_{1} - \theta_{2}} \right)} + {\sin \left( {\theta_{2} - \theta_{3}} \right)} + {\sin \left( {\theta_{3} - \theta_{1}} \right)}}.}} & (5)\end{matrix}$

The inverse shows that conditions exist where a unique solution for theloads might not be possible. Specifically, the conditions exist wherethe inverse is singular, as in the condition in equation 6.

sin(θ₁−θ₂)+sin(θ₂−θ₃)+sin(θ_(3−θ) ₁)=0  (6)

Notice that the selection of a reference axis (azimuth of 0) isarbitrary. Rotating the axis to a new position has no effect on any ofthe sine terms since they depend only on the difference in angles.Consequently, θ₃ can be set to 0 with no loss of generality. Thesingularity constraint can be restated as in equation 7. $\begin{matrix}{{{{\sin \left( {\theta_{1} - \theta_{2}} \right)} + {\sin \left( \theta_{2} \right)} - {\sin \left( \theta_{1} \right)}} = 0}{{{\sin \quad \theta_{1}\cos \quad \theta_{2}} - {\sin \quad \theta_{2}\cos \quad \theta_{1}} + {\sin \quad \theta_{2}} - {\sin \quad \theta_{1}}} = 0}{\frac{\sin \quad \theta_{2}}{1 - {\cos \quad \theta_{2}}} = \frac{\sin \quad \theta_{1}}{1 - {\cos \quad \theta_{1}}}}{{\cot \frac{\theta_{2}}{2}} = {\cot \frac{\theta_{1}}{2}}}} & (7)\end{matrix}$

The final relationship is satisfied only when θ₁=θ₂. Accordingly, theload components can be determined as long as no two gauges have the samegauge angle and the same azimuth.

Solving equations 3 and 4 results in equations 8 and 9. Therelationships are completely general for a six axis load cell consistingof three gauges places at zero gauge angle and three gauges placed atfir Any set of three unique azimuth positions can be chosen for each setof three equations. $\begin{matrix}{\begin{bmatrix}\begin{matrix}P_{x} \\P_{y}\end{matrix} \\M_{z}\end{bmatrix} = {\frac{E\begin{bmatrix}{{- \frac{D - d}{Q_{\max}}}\left( {{\cos \quad \theta_{2}} - {\cos \quad \theta_{3}}} \right)} & {{- \frac{D - d}{Q_{\max}}}\left( {{\cos \quad \theta_{3}} - {\cos \quad \theta_{1}}} \right)} & {{- \frac{D - d}{Q_{\max}}}\left( {{\cos \quad \theta_{1}} - {\cos \quad \theta_{2}}} \right)} \\{\frac{D - d}{Q_{\max}}\left( {{\sin \quad \theta_{3}} - {\sin \quad \theta_{2}}} \right)} & {\frac{D - d}{Q_{\max}}\left( {{\sin \quad \theta_{1}} - {\sin \quad \theta_{3}}} \right)} & {\frac{D - d}{Q_{\max}}\left( {{\sin \quad \theta_{2}} - {\sin \quad \theta_{1}}} \right)} \\{\frac{J}{r}{\sin \left( {\theta_{2} - \theta_{3}} \right)}} & {\frac{J}{r}{\sin \left( {\theta_{3} - \theta_{1}} \right)}} & {\frac{J}{r}{\sin \left( {\theta_{1} - \theta_{2}} \right)}}\end{bmatrix}}{2\sqrt{v}\left( {{\sin \left( {\theta_{1} - \theta_{2}} \right)} + {\sin \left( {\theta_{2} - \theta_{3}} \right)} + {\sin \left( {\theta_{3} - \theta_{1}} \right)}} \right)}\begin{bmatrix}\begin{matrix}{ɛ\left( {\beta_{\tau},\theta_{1}} \right)} \\{ɛ\left( {\beta_{\tau},\theta_{2}} \right)}\end{matrix} \\{ɛ\left( {\beta_{\tau},\theta_{3}} \right)}\end{bmatrix}}} & (8) \\{\begin{bmatrix}\begin{matrix}M_{x} \\M_{y}\end{matrix} \\P_{z}\end{bmatrix} = {\frac{E\begin{bmatrix}{\frac{I}{R}\left( {{\cos \quad \theta_{2}} - {\cos \quad \theta_{3}}} \right)} & {\frac{I}{R}\left( {{\cos \quad \theta_{3}} - {\cos \quad \theta_{1}}} \right)} & {\frac{I}{R}\left( {{\cos \quad \theta_{1}} - {\cos \quad \theta_{2}}} \right)} \\{{- \frac{I}{R}}\left( {{\sin \quad \theta_{3}} - {\sin \quad \theta_{2}}} \right)} & {{- \frac{I}{R}}\left( {{\sin \quad \theta_{1}} - {\sin \quad \theta_{3}}} \right)} & {{- \frac{I}{R}}\left( {{\sin \quad \theta_{2}} - {\sin \quad \theta_{1}}} \right)} \\{A\quad {\sin \left( {\theta_{2} - \theta_{3}} \right)}} & {A\quad {\sin \left( {\theta_{3} - \theta_{1}} \right)}} & {A\quad {\sin \left( {\theta_{1} - \theta_{2}} \right)}}\end{bmatrix}}{\left( {{\sin \left( {\theta_{1} - \theta_{2}} \right)} + {\sin \left( {\theta_{2} - \theta_{3}} \right)} + {\sin \left( {\theta_{3} - \theta_{1}} \right)}} \right)}\begin{bmatrix}\begin{matrix}{ɛ\left( {0,\theta_{1}} \right)} \\{ɛ\left( {0,\theta_{2}} \right)}\end{matrix} \\{ɛ\left( {0,\theta_{3}} \right)}\end{bmatrix}}} & (9)\end{matrix}$

A load cell design can comprise gauges mounted every 60 degrees aboutthe circumference of a cylinder. If the zero angle gauges are mounted at0, 120, and 240 degrees and the β_(τ) angle gauges are mounted at 60,180, and 300 degrees, then the resulting load components are given byequation 10. $\begin{matrix}{{P_{x} = {- {\frac{E\left( {D - d} \right)}{2\sqrt{3v}Q_{\max}}\left\lbrack {{ɛ\left( {\beta_{\tau},\theta_{2}} \right)} - {ɛ\left( {\beta_{\tau},\theta_{3}} \right)}} \right\rbrack}}}{P_{y} = {- {\frac{E\left( {D - d} \right)}{6\sqrt{v}Q_{\max}}\left\lbrack {{2{ɛ\left( {\beta_{\tau},\theta_{1}} \right)}} - {ɛ\left( {\beta_{\tau},\theta_{2}} \right)} - {ɛ\left( {\beta_{\tau},\theta_{3}} \right)}} \right\rbrack}}}{P_{z} = {\frac{EA}{3}\left\lbrack {{ɛ\left( {0,\theta_{1}} \right)} + {ɛ\left( {0,\theta_{2}} \right)} + {ɛ\left( {0,\theta_{3}} \right)}} \right\rbrack}}{M_{x} = {\frac{EI}{\sqrt{3}r}\left\lbrack {{ɛ\left( {0,\theta_{2}} \right)} - {ɛ\left( {0,\theta_{3}} \right)}} \right\rbrack}}{M_{y} = {\frac{EI}{3r}\left\lbrack {{2{ɛ\left( {0,\theta_{1}} \right)}} - {ɛ\left( {0,\theta_{2}} \right)} - {ɛ\left( {0,\theta_{3}} \right)}} \right\rbrack}}{M_{z} = {\frac{J}{6\sqrt{v}r}\left\lbrack {{ɛ\left( {\beta_{\tau},\theta_{1}} \right)} + {ɛ\left( {\beta_{\tau},\theta_{2}} \right)} + {ɛ\left( {\beta_{\tau},\theta_{3}} \right)}} \right\rbrack}}} & (10)\end{matrix}$

The surface curvature of a circular cross section load cell can posesproblems for small radius load cells. A polygonal shape can be used withthe present invention, allowing each strain gauge to mount with a flatsurface. The stiffness properties of the polygonal shape are important,specifically the polar moment of inertia. As an example, a square crosssection could be used. One axial (0 degrees) and one β_(τ) gauge can bemounted with each of three faces (azimuths of 0, +90, and −90 degrees).The corresponding relations are given in equation 11. $\begin{matrix}{{P_{x} = {- {\frac{E\left( {D - d} \right)}{4\sqrt{v}Q_{\max}}\left\lbrack {{ɛ\left( {\beta_{\tau},\theta_{2}} \right)} - {ɛ\left( {\beta_{\tau},\theta_{3}} \right)}} \right\rbrack}}}{P_{y} = {- {\frac{E\left( {D - d} \right)}{4\sqrt{v}Q_{\max}}\left\lbrack {{2{ɛ\left( {\beta_{\tau},\theta_{1}} \right)}} - {ɛ\left( {\beta_{\tau},\theta_{2}} \right)} - {ɛ\left( {\beta_{\tau},\theta_{3}} \right)}} \right\rbrack}}}{P_{z} = {\frac{EA}{2}\left\lbrack {{ɛ\left( {0,\theta_{2}} \right)} + {ɛ\left( {0,\theta_{3}} \right)}} \right\rbrack}}{M_{x} = {{{\frac{EI}{2r}\left\lbrack {{ɛ\left( {0,\theta_{2}} \right)} - {ɛ\left( {0,\theta_{3}} \right)}} \right\rbrack}M_{y}} = {{{\frac{EI}{2r}\left\lbrack {{2{ɛ\left( {0,\theta_{1}} \right)}} - {ɛ\left( {0,\theta_{2}} \right)} - {ɛ\left( {0,\theta_{3}} \right)}} \right\rbrack}M_{z}} = {\frac{5J}{32\sqrt{v}r}\left\lbrack {{ɛ\left( {\beta_{\tau},\theta_{2}} \right)} + {ɛ\left( {\beta_{\tau},\theta_{3}} \right)}} \right\rbrack}}}}} & (11)\end{matrix}$

Dummy resistors can be used in combination with the strain gauges toproduce bridges to facilitate determination of strain gauge outputs. Thecombination of strain gauge outputs can be accomplished, for example,with analog electronic circuitry or with well-known microprocessors.

The particular sizes and equipment discussed above are cited merely toillustrate particular embodiments of the invention. It is contemplatedthat the use of the invention may involve components having differentsizes and characteristics. It is intended that the scope of theinvention be defined by the claims appended hereto.

We claim:
 1. A load cell comprising: a) a body having a body axis, firstand second ends, made of a material having a Poisson's ratio (v); b)first, second, and third strain gauges mounted with the body at azimuthsdifferent from each other and at orientations relative to the axis otherthan, 0 and 90 degrees; and c) fourth, fifth, and sixth strain gaugesmounted with the body at azimuths different from each other and atorientations other than substantially$\frac{1}{2}{\arccos \left( \frac{v - 1}{v + 1} \right)}$

and substantially${- \frac{1}{2}}{{\arccos \left( \frac{v - 1}{v + 1} \right)}.}$


2. A load cell according to claim 1, wherein the first, second, andthird strain gauges are mounted at orientations of substantially$\frac{1}{2}{\arccos \left( \frac{v - 1}{v + 1} \right)}$

or substantially${- \frac{1}{2}}{{\arccos \left( \frac{v - 1}{v + 1} \right)}.}$


3. A load cell according to claim 1, wherein the fourth, fifth, andsixth strain gauges are mounted at orientations of substantially 0degrees or substantially 90 degrees.
 4. A load cell according to claim1, wherein the body has a substantially circular cross section, andwherein the plurality of strain gauges are mounted at substantially evenazimuthal intervals about a circumference of the circle.
 5. A load cellaccording to claim 1, wherein the body has a substantially rectangularcross section, and wherein the first, second, and third strain gaugesare mounted with the body at locations corresponding to different facesof the rectangular cross section.
 6. A load cell according to claim 1,wherein the body has a substantially polygonal cross section, andwherein the first, second, and third strain gauges are mounted with thebody at locations corresponding to different faces of the polygonalcross section.
 7. A load cell according to claim 6, wherein the body hasa substantially rectangular cross section, and wherein the fourth,fifth, and sixth strain gauges are mounted with the body at locationscorresponding to different faces of the rectangular cross section.
 8. Aload cell comprising: a) a body having a body axis, first and secondends, adapted to receive a load applied made of a material having aPoisson's ratio (v); and b) first, second, and third strain gaugesmounted with the body at azimuths different from each other and atorientations relative to the axis of substantially$\frac{1}{2}{\arccos \left( \frac{v - 1}{v + 1} \right)}$

or substantially${- \frac{1}{2}}{{\arccos \left( \frac{v - 1}{v + 1} \right)}.}$


9. A load cell according to claim 8, wherein the body has asubstantially circular cross section, and wherein the first, second, andthird strain gauges are mounted at substantially even azimuthalintervals about a circumference of the circle.
 10. A load cell accordingto claim 8, wherein the body has a substantially rectangular crosssection, and wherein the first, second, and third strain gauges aremounted with the body at locations corresponding to different faces ofthe rectangular cross section.
 11. A load cell according to claim 8,wherein the body has a substantially polygonal cross section, andwherein the first, second, and third strain gauges are mounted with thebody at locations corresponding to different faces of the polygonalcross section.
 12. A load cell comprising: a) a body having a body axis,first and second ends, made of a material having a Poisson's ratio (v);and b) first, second, and third strain gauges mounted with the body atazimuths different from each other and at orientations relative to theaxis of substantially 0 degrees or substantially 90 degrees.
 13. A loadcell according to claim 12, wherein the body has a substantiallycircular cross section, and wherein the first, second, and third straingauges are mounted at substantially even azimuthal intervals about acircumference of the circle.
 14. A load cell according to claim 12,wherein the body has a substantially rectangular cross section, andwherein the first, second, and third strain gauges are mounted with thebody at locations corresponding to different faces of the rectangularcross section.
 15. A load cell according to claim 12, wherein the bodyhas a substantially polygonal cross section, and wherein the first,second, and third strain gauges are mounted with the body at locationscorresponding to different faces of polygonal cross section.